In Fig. REF , we show the results of \(\kappa _{c, {\rm eff}}\)  for \(N_t=6\)  as function of \(n_{\rm max}\) .
Black and red data are obtained using \(\kappa _{c, {\rm LO}}\)  on \(24^3 \times 6\)  and \(32^3 \times 6\)  lattices, respectively.
Our result \(\kappa _{c, {\rm eff}} = 0.1205(23)\)  for \(n_{\rm max}=N_t+2\)  on the \(24^3 \times 6\)  lattice is consistent with the result of the next-to-leading-order calculation given in Ref. [1]}.
As \(n_{\rm max}\)  increases, \(\kappa _{c, {\rm eff}}\)  converges to
\(\kappa _{c, {\rm eff}} = 0.1134(18)\)  for the \(24^3 \times 6\)  lattice and
\(0.1090(25)\)  for \(32^3 \times 6\) .
We find that the hopping parameter expansion works well for \(N_t=6\)  when high order effects are incorporated up to a sufficiently high order.
At the same time, we note sizable finite volume effect in \(\kappa _{c, {\rm eff}}\) .
Large finite volume effect was reported also in the leading-order determination of the critical point by the shape of the histogram of the Polyakov loop [2]}.
It is important to perform a systematic study with large spatial volumes to remove the finite volume effect.
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